Elements of Algebra

CHAP. VII. ADDITIONS. 575 $m' = \frac{1}{-1} = -1$, $n'' = 1 - 1 = 0$, $D''' = \frac{-13}{-1} = 13$, $y' = -y'' + y'''$. Thus, we have the same values of $D''$, $D'''$, and $n''$, as before; so that the transformed equation in $y''$, and $y'''$, will likewise be the same. We shall, therefore, have also $y''' = 18$, and $y'' = 5$; wherefore, $y' = -y'' + y''' = 13$, and $y = -3y' + y'' = -34$. So that we have found two values of $y$, with the corresponding values of $y'$, or $z$; and these values result from the supposition of $n = \mp 35$. Now, as we cannot find any other value of $n$, with the requisite conditions, it follows that the preceding values will be the only primitive values that we can have; but we may then find from them an infinite number of derivative values by the method of Art. 72. Taking, therefore, these values of $y$ and $z$ for $p$ and $q$, we shall have, in general, by the same Article, $y = 74t - (101 \times 23 - 35 \times 74)u = 74t + 267u$ $z = 23t + (12 \times 74 - 35 \times 23)u = 23t + 83u$; or $y = -34t - (101 \times 13 - 35 \times 34)u = -34t - 123u$ $z = 13t + (-12 \times 34 + 35 \times 13)u = 13t + 47u$; and we shall only have farther to deduce the values of $t$ and $u$ from the equation, $t^2 - 13u^2 = 1$. Now, all these values may be found already calculated in the Table at the end of Chap. VII. of the preceding Treatise: we shall therefore immediately have $t = 649$, and $u = 180$; so that taking these values for $T$ and $v$, in the formulæ of Art. 75, we shall have, in general, $t = \frac{(649+180\sqrt{13})^m+(649-180\sqrt{13})^m}{2}$, $u = \frac{(649+180\sqrt{13})^m-(649-180\sqrt{13})^m}{2\sqrt{13}}$; where we may give to $m$ whatever value we choose, provided we take only positive whole numbers. Now, as the values of $t$ and $u$ may be taken both positive and negative, the values of $y$, which satisfy the question, will all be contained in these two formulæ, $y = \pm 74t \pm 267u$, and $y = \pm 34t \pm 123u$, the doubtful signs being arbitrary.